Why did Skylab get hot and Apollo 13 get cold when they both lost power. I've heard that the Skylab interior got very warm (120 deg.+) when it lost a solar panel, and yet Apollo 13 froze inside when it lost power from its fuel cells. It seem that whatever caused the problem in one could have helped the other. What gives?
The issue was partly electric power (which is actually small potatoes in the thermal economy of most spacecraft) but also the balance of radiant power received and lost by the two different vehicles in two somewhat different situations. (Actually, for Skylab, although a solar panel was indeed lost, the most important thing was that the entire thermal control surface surrounding the habitat was ripped off.) The whole subject of why things in space are hot or cold was actually just recently discussed in another MAD Scientist answer, so for more details you might want to look there.
However, very briefly, the temperature of a passive body (ie, a body not generating much energy of its own) in space is determined by the balance of heat received by radiation from the environment, and heat radiated away to space by the body. If these are not equal, the body will either warm up and begin to radiate more power (if more power is coming in than being lost), or cool off and radiate less (if more is being radiated than received). In either case, eventually the body will warm up or cool off to the point that the heat going out just equals the heat coming in, and that will determine its final temperature. (All this assumes that the environment itself and the properties of the body are not changing enough to matter, of course -- the situation has to be static in that sense, or else it will never achieve a constant temperature.) Thus it is kind of a matter of bookkeeping: adding up all the heat energy received from other bodies in space, and then figuring out the temperature the body has to have to radiate just that much power away. This is a fussy calculation, and I will not try to give a quantitative answer here. However, there are two major factors that would have to be considered.
The first is that Apollo 13 was far enough out in deep space that it received power from the Sun (about 1.38 kW per m²) but relatively little from the Earth. Since the Earth must on the average radiate as much energy as it receives from the Sun (why?!), the Earth's contribution to the radiant power in its immediate vicinity is not negligible at all, although much of the energy escapes in the infrared. Of course we must also remember that the Earth blocks the radiation received from the Sun during about 1/3 of each orbit. In any event, the environment half-way between the Earth and the Moon is actually significantly different than that immediately adjacent to the Earth, where Skylab's orbit carried it, and this is one of the two big factors a proper calculation would have to consider.
The other factor is the loss of Skylab's thermal control shield, which was specifically designed to adjust its absorption of sunlight and infrared from the Earth and its emission of infrared radiation to yield the desired temperature in the low-earth environment. It is not surprising that the loss of the shield affected the temperature and made it different, either too hot or too cold. A body in space can (when the radiative calculation is done) be quite hot or quite cold, so I would just have to say that by accident it turned out that Skylab without its shield was too hot.
There was another important factor for Apollo 13, and that (oddly enough) had to do with water. Water was used to cool the electronics, and of all the consumables in short supply during the Apollo 13 emergency, water was perhaps the worst situation. In order to reduce the requirement for cooling water to prevent the electronics from overheating, the equipment in the Lunar Module was largely powered off. Of course the Command Module was powered down completely because of the primary failure, which shut down the fuel cells. Although the internal heat is typically a fairly small part of the total thermal budget of a spacecraft (just compare that 1.38 kW per m², totaled over the spacecraft area, with the typical electric power available in order to see this), it does enter in and does have to be considered. With the electronics powered off, the heat it generated normally was not present, and that is the main reason Apollo 13 got so cold during trans-Earth coast. In that sense it was like Skylab: the system was designed for one set of conditions, and the failures put both spacecraft into situations for which the engineers had not, could not, design in advance.